.25n(6n-3)=4n+1

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Solution for .25n(6n-3)=4n+1 equation:


Simplifying
0.25n(6n + -3) = 4n + 1

Reorder the terms:
0.25n(-3 + 6n) = 4n + 1
(-3 * 0.25n + 6n * 0.25n) = 4n + 1
(-0.75n + 1.5n2) = 4n + 1

Reorder the terms:
-0.75n + 1.5n2 = 1 + 4n

Solving
-0.75n + 1.5n2 = 1 + 4n

Solving for variable 'n'.

Reorder the terms:
-1 + -0.75n + -4n + 1.5n2 = 1 + 4n + -1 + -4n

Combine like terms: -0.75n + -4n = -4.75n
-1 + -4.75n + 1.5n2 = 1 + 4n + -1 + -4n

Reorder the terms:
-1 + -4.75n + 1.5n2 = 1 + -1 + 4n + -4n

Combine like terms: 1 + -1 = 0
-1 + -4.75n + 1.5n2 = 0 + 4n + -4n
-1 + -4.75n + 1.5n2 = 4n + -4n

Combine like terms: 4n + -4n = 0
-1 + -4.75n + 1.5n2 = 0

Begin completing the square.  Divide all terms by
1.5 the coefficient of the squared term: 

Divide each side by '1.5'.
-0.6666666667 + -3.166666667n + n2 = 0

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + -3.166666667n + 0.6666666667 + n2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + -3.166666667n + n2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + -3.166666667n + n2 = 0 + 0.6666666667
-3.166666667n + n2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
-3.166666667n + n2 = 0.6666666667

The n term is -3.166666667n.  Take half its coefficient (-1.583333334).
Square it (2.506944447) and add it to both sides.

Add '2.506944447' to each side of the equation.
-3.166666667n + 2.506944447 + n2 = 0.6666666667 + 2.506944447

Reorder the terms:
2.506944447 + -3.166666667n + n2 = 0.6666666667 + 2.506944447

Combine like terms: 0.6666666667 + 2.506944447 = 3.1736111137
2.506944447 + -3.166666667n + n2 = 3.1736111137

Factor a perfect square on the left side:
(n + -1.583333334)(n + -1.583333334) = 3.1736111137

Calculate the square root of the right side: 1.781463195

Break this problem into two subproblems by setting 
(n + -1.583333334) equal to 1.781463195 and -1.781463195.

Subproblem 1

n + -1.583333334 = 1.781463195 Simplifying n + -1.583333334 = 1.781463195 Reorder the terms: -1.583333334 + n = 1.781463195 Solving -1.583333334 + n = 1.781463195 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.583333334' to each side of the equation. -1.583333334 + 1.583333334 + n = 1.781463195 + 1.583333334 Combine like terms: -1.583333334 + 1.583333334 = 0.000000000 0.000000000 + n = 1.781463195 + 1.583333334 n = 1.781463195 + 1.583333334 Combine like terms: 1.781463195 + 1.583333334 = 3.364796529 n = 3.364796529 Simplifying n = 3.364796529

Subproblem 2

n + -1.583333334 = -1.781463195 Simplifying n + -1.583333334 = -1.781463195 Reorder the terms: -1.583333334 + n = -1.781463195 Solving -1.583333334 + n = -1.781463195 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.583333334' to each side of the equation. -1.583333334 + 1.583333334 + n = -1.781463195 + 1.583333334 Combine like terms: -1.583333334 + 1.583333334 = 0.000000000 0.000000000 + n = -1.781463195 + 1.583333334 n = -1.781463195 + 1.583333334 Combine like terms: -1.781463195 + 1.583333334 = -0.198129861 n = -0.198129861 Simplifying n = -0.198129861

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.364796529, -0.198129861}

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